Search results for "Q55"
showing 10 items of 34 documents
Real quadrics in C n , complex manifolds and convex polytopes
2006
In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics Cn which are invariant with respect to the natural action of the real torus (S1)n onto Cn. The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-cros…
On critical behaviour in systems of Hamiltonian partial differential equations
2013
Abstract We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (P $$_I$$ I ) equation or its fourth-order analogue P $$_I^2$$ I 2 . As concrete examples, we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture.
Families of rational solutions to the KPI equation of order 7 depending on 12 parameters
2017
International audience; We construct in this paper, rational solutions as a quotient of two determinants of order 2N = 14 and we obtain what we call solutions of order N = 7 to the Kadomtsev-Petviashvili equation (KPI) as a quotient of 2 polynomials of degree 112 in x, y and t depending on 12 parameters. The maximum of modulus of these solutions at order 7 is equal to 2(2N + 1)2= 450. We make the study of the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6. When all these parameters grow, triangle and ring structures are obtained.
Rational solutions to the KdV equation depending on multi-parameters
2021
We construct multi-parametric rational solutions to the KdV equation. For this, we use solutions in terms of exponentials depending on several parameters and take a limit when one of these parameters goes to 0. Here we present degenerate rational solutions and give a result without the presence of a limit as a quotient of polynomials depending on 3N parameters. We give the explicit expressions of some of these rational solutions.
Higher order Peregrine breathers solutions to the NLS equation
2015
The solutions to the one dimensional focusing nonlinear Schrödinger equation (NLS) can be written as a product of an exponential depending on t by a quotient of two polynomials of degree N (N + 1) in x and t. These solutions depend on 2N − 2 parameters : when all these parameters are equal to 0, we obtain the famous Peregrine breathers which we call PN breathers. Between all quasi-rational solutions of the rank N fixed by the condition that its absolute value tends to 1 at infinity and its highest maximum is located at the point (x = 0, t = 0), the PN breather is distinguished by the fact that PN (0, 0) = 2N + 1. We construct Peregrine breathers of the rank N explicitly for N ≤ 11. We give …
Hierarchy of solutions to the NLS equation and multi-rogue waves.
2014
The solutions to the one dimensional focusing nonlinear Schrödinger equation (NLS) are given in terms of determinants. The orders of these determinants are arbitrarily equal to 2N for any nonnegative integer $N$ and generate a hierarchy of solutions which can be written as a product of an exponential depending on t by a quotient of two polynomials of degree N(N+1) in x and t. These solutions depend on 2N-2 parameters and can be seen as deformations with 2N-2 parameters of the Peregrine breather P_{N} : when all these parameters are equal to 0, we recover the P_{N} breather whose the maximum of the module is equal to 2N+1. Several conjectures about the structure of the solutions are given.
Families of deformations of the thirteen peregrine breather solutions to the NLS equation depending on twenty four parameters
2017
International audience; We go on with the study of the solutions to the focusing one dimensional nonlinear Schrodinger equation (NLS). We construct here the thirteen's Peregrine breather (P13 breather) with its twenty four real parameters, creating deformation solutions to the NLS equation. New families of quasirational solutions to the NLS equation in terms of explicit ratios of polynomials of degree 182 in x and t multiplied by an exponential depending on t are obtained. We present characteristic patterns of the modulus of these solutions in the (x; t) plane, in function of the different parameters.
Families of solutions of order nine to the NLS equation with sixteen parameters
2015
We construct new deformations of the Peregrine breather (P9) of order 9 with 16 real parameters. With this method, we obtain explicitly new families of quasi-rational solutions to the NLS equation in terms of a product of an exponential depending on t by a ratio of two polynomials of degree 90 in x and t; when all the parameters are equal to 0, we recover the classical P9 breather. We construct new patterns of different types of rogue waves as triangular configurations of 45 peaks as well as rings and concentric rings.
2N+1 highest amplitude of the modulus of the N-th order AP breather and other 2N-2 parameters solutions to the NLS equation
2015
We construct here new deformations of the AP breather (Akhmediev-Peregrine breather) of order N (or AP N breather) with 2N −2 real parameters. Other families of quasi-rational solutions of the NLS equation are obtained. We evaluate the highest amplitude of the modulus of AP breather of order N ; we give the proof that the highest amplitude of the AP N breather is equal to 2N + 1. We get new formulas for the solutions of the NLS equation, different from these already given in previous works. New solutions for the order 8 and their deformations according to the parameters are explicitly given. We get the triangular configurations as well as isolated rings at the same time. Moreover, the appea…
Families of solutions to the CKP equation with multi-parameters
2020
We construct solutions to the CKP (cylindrical Kadomtsev-Petviashvili)) equation in terms of Fredholm determinants. We deduce solutions written as a quotient of wronskians of order 2N. These solutions are called solutions of order N ; they depend on 2N − 1 parameters. They can be written as a quotient of 2 polynomials of degree 2N (N + 1) in x, t and 4N (N + 1) in y depending on 2N − 2 parameters. We explicitly construct the expressions up to order 5 and we study the patterns of their modulus in plane (x, y) and their evolution according to time and parameters.